Natural Numbers
All the positive integers from 1, 2, 3,……, ∞.
Whole Numbers
All the natural numbers including zero are called Whole Numbers.
Integers
All negative and positive numbers including zero are called Integers.
Rational Numbers
A number is called Rational if it can be expressed in the form p/q where p and q are integers
(q> 0). It includes all natural, whole number and integers.Example: 1/2, 4/3, 5/7,1 etc.
1.2 PROPERTIES OF RATIONAL NUMBERS:
 Closure Property
This shows that the operation of any two same types of numbers is also the same type or not.
a). Whole Numbers
If p and q are two whole numbers then
Operation  Addition  Subtraction  Multiplication  Division 
Whole number 
p + q will also be the whole number. 
p – q will not always be a whole number. 
pq will also be the whole number. 
p ÷ q will not always be a whole number. 
Example 
6 + 0 = 6 
8 – 10 = – 2 
3 × 5 = 15 
3 ÷ 5 = 3/5 
Closed or Not 
Closed 
Not closed 
Closed 
Not closed 
b). Integers
If p and q are two integers then
Operation 
Addition 
Subtraction 
Multiplication 
Division 
Integers 
p+q will also be an integer. 
pq will also be an integer. 
pq will also be an integer. 
p ÷ q will not always be an integer. 
Example   3 + 2 = – 1  5 – 7 = – 2   5 × 8 = – 40   5 ÷ 7 = – 5/7 
Closed or not  Closed  Closed  Closed  Not closed 
c). Rational Numbers
If p and q are two rational numbers then
Operation 
Addition 
Subtraction 
Multiplication 
Division 
Rational Numbers  p + q will also be a rational number.  p – q will also be a rational number.  pq will also be a rational number.  p ÷ q will not always be a rational number 
Example  p ÷ 0
= not defined 

Closed or not 
Closed 
Closed 
Closed 
Not closed 
 Commutative Property
This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.
a). Whole Numbers
If p and q are two whole numbers then
Operation  Addition  Subtraction  Multiplication  Division 
Whole number  p + q = q + p  p – q ≠ q – p  p × q = q × p  p ÷ q ≠ q ÷ p 
Example  3 + 2 = 2 + 3  8 –10 ≠ 10 – 8 – 2 ≠ 2  3 × 5 = 5 × 3  3 ÷ 5 ≠ 5 ÷ 3 
Commutative  yes  No  yes  No 
b). Integers
If p and q are two integers then
Operation  Addition  Subtraction  Multiplication  Division 
Whole number  p + q = q + p  p – q ≠ q – p  p × q = q × p  p ÷ q ≠ q ÷ p 
Example  True  5 – 7 = – 7 – (5)   5 × 8 = 8 × (–5)   5 ÷ 7 ≠ 7 ÷ (5) 
Commutative  yes  No  yes  No 
c). Rational Numbers
If p and q are two rational numbers then
Operation  Addition  Subtraction  Multiplication  Division 
Example 
= + (

( ) 

=


Example  True  5 – 7 = – 7 – (5)   5 × 8 = 8 × (–5)   5 ÷ 7 ≠ 7 ÷ (5) 
Commutative  yes  No  yes  No 
 Associative Property
This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.
a). Whole Numbers
If p, q and r are three whole numbers then
Operation  Addition  Subtraction  Multiplication  Division 
Whole number  p + (q + r) = (p + q) + r  p – (q – r) = (p – q) – r  p × (q × r) = (p × q) × r  p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r 
Example  3 + (2 + 5) = (3 + 2) + 5  8 – (10 – 2) ≠ (8 10) – 2  3 × (5 × 2) = (3 × 5) × 2  10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1 
Associative  yes  No  yes  No 
 Integers
If p, q and r are three integers then
Operation  Integers  Example  Associative 
Addition  p + (q + r) = (p + q) + r  (– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5)  Yes 
Subtraction  p – (q – r) = (p – q) – r  5 – (7 – 3) ≠ (5 – 7) – 3  No 
Multiplication  p × (q × r) = (p × q) × r  (– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5)  Yes 
Division  p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r

[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)]  No 
 Rational Numbers
If p, q and r are three rational numbers then
Operation  Integers  Example  Associative 
Addition  p + (q + r) = (p + q) + r  =
= =

Yes 
Subtraction  p – (q – r) = (p – q) – r  No  
Multiplication  p × (q × r) = (p × q) × r  = ×
Hence LHS = RHS 
Yes 
Division  p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r 

No 
The Role of Zero in Numbers (Additive Identity)
Zero is the additive identity for whole numbers, integers and rational numbers.
Identity  Example  
Whole number  a + 0 = 0 + a = a  Addition of zero to whole number  2 + 0 = 0 + 2 = 2 
Integer  b + 0 = 0 + b = b  Addition of zero to an integer  False 
Rational number  c + 0 = 0 + c = c  Addition of zero to a rational number  2/5 + 0 = 0 + 2/5 = 2/5 
The Role of one in Numbers (Multiplicative Identity)
One is the multiplicative identity for whole numbers, integers and rational numbers.
Identity  Example  
Whole number  a ×1 = a  Multiplication of one to the whole number  5 × 1 = 5  
Integer  b × 1= b  Multiplication of one to an integer   5 × 1 = – 5  
Rational Number  c × 1= c  Multiplication of one to a rational number 


Negative of a Number (Additive Inverse)
Identity  Example  
Whole number  a +( a) = 0  Where a is a whole number  5 + (5) = 0 
Integer  b +( b) = 0  Where b is an integer  True 
Rational number  c + (c) = 0  Where c is a rational number 

Reciprocal (Multiplicative Inverse)
The multiplicative inverse of any rational number
Example :The reciprocal of is.
Distributivity of Multiplication over Addition and Subtraction for Rational Numbers
This shows that for all rational numbers p, q and r
 p(q + r) = pq + pr2. p(q – r) = pq – pr
Example:Check the distributive property of the three rational numbers 4/7,( 2)/3 and 1/2.
Solution:Let’s find the value of